Assume the random variable x is normally distributed with mean muequals50 and standard deviation sigmaequals7. compute the probability. be sure to draw a normal curve with the area corresponding to the probability shaded. p(55less than or equalsxless than or equals70)

P [ 55 ≤ X ≤ 70 ] = 23.55 %

Step-by-step explanation:

We know:

μ₀ = mean of population              σ  = standard deviation  

We begin for calculating the probability of X ≤ 55 so

Step 1:

μ₀  = 50              σ = 7

×₁    =   ( μ - μ₀ ) ÷ σ   ⇒  ×₁  = (55-50) ÷ 7     ⇒  ×₁  = 5 ÷ 7    ×₁ = 0.7143

We look z table and we have to interpolate

value (closest smaller Than    ×₁   0.71   and closest bigger than ×₁   0.72

The associated area for these point are:   0.7611 and 0.7642

Taking diferences and by rule of three

0.01             ⇒  0.0031

0.0042        ⇒      Δ          Δ = 0.0013     and  Area for ×₁  = 0.7624

That is the probability of  X ≤ 55

Now we have to find the probability of X ≤ 70

We procede as in step 1

μ₀  = 50             σ = 7

×₂    =   ( μ - μ₀ ) ÷ σ   ⇒  ×₂  = (70-50) ÷ 7     ⇒  ×₂  = 20 ÷ 7    ×₂ = 2.8571

We look z table and we have to interpolate

for 2.85    ⇒ 0.9978 (area)              2.8571 - 2.85  = 0.007

for 2.86    ⇒ 0.9979 (area)

Therefore by rule of three we have

0.01      ⇒    0.0001

0.007   ⇒        Δ                       Δ = 0.00007

And area for      ×₂ = 2.8571      is  equal to 0.99787   and the probability of

X ≤ 0.99787   or    99.79

Now if we look the annex drawing (the area we are looking for is enclose for the values x₁   and ×₂  (pink area) and is the diference between these two areas

So   P [ 55 ≤ X ≤ 70 ]  is 99.79 % - 76.24 %  = 23.55

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